Nonlinear optics plays a central role in the advancement of optical science and laser‐based technologies. The second‐order rogue‐wave solutions and modulation instability for the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling in nonlinear optics are reported in this paper. Generalized Darboux transformations for such coupled equations are derived, with which the second‐order rational solutions for the purpose of modelling the rogue waves are obtained. With respect to the slowly‐varying complex amplitudes of two interacting optical modes, it is observed that 1) number of valleys of the second‐order rogue waves increases and peak value of the second‐order rogue wave decreases first and then increases; 2) single‐hump second‐order rogue wave turns into the double‐hump second‐order rogue wave; 3) single‐hump bright second‐order rogue wave turns into the dark second‐order rogue wave and finally becomes the three‐hump bright second‐order rogue wave. Meanwhile, baseband modulation instability through the linear stability analysis is seen. The second‐order rogue‐wave solutions and modulation instability for the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling are reported. With respect to the slowly‐varying complex amplitudes of two interacting optical modes, different types of rogue structures are presented and properties of the rogue‐wave solutions are discussed.